Restrictions on wave equations for passive media

Abstract

Most derivations of acoustic wave equations involve ensuring that causality is satisfied. Here, the consequences of also requiring that the medium should be passive are explored. This is a stricter criterion than causality for a linear system and implies that there are restrictions on the relaxation modulus and its first few derivatives. The viscous and relaxation models of acoustics satisfy passivity and have restrictions on not only a few, but all derivatives of the relaxation modulus. These models are described as a system of springs and dampers with positive parameters and belong to the important class of completely monotone systems. It is shown here that the attenuation as a function of frequency for such media has to increase slower than a linear function. Likewise, the phase velocity has to increase monotonically. This gives criteria on which one may judge whether a proposed wave equation is passive or not, as illustrated by comparing two different versions of the viscous wave equation.

Towards a consensus on rheological models for elastography in soft tissues

“…The complex wave number in linear wave propagation is (Holm and Holm, 2017): 20) where k has a real and imaginary part ki   as described in more detail in Appendix 1. Each of the classical and multiple relaxation models considered in the previous sections prescribe a frequency dependence of the real and imaginary parts of   , thus determining the complex behavior of wavenumber, k .…”

Distributed-order fractional constitutive stress–strain relation in wave propagation modeling

“…Considerations regarding the different forms of wave equations for the viscoelastic media can be found in [10,23,27,28,30,31,35,36,38,53,55], including derivation and solution of a model describing propagation and attenuation of two-dimensional dilatational waves; derivation of thermodynamically consistent fractional wave equation which allows for the calculation of LAMB waves; analysis of vibrations damping; proving the existence of global solution to memory-type wave equation; application of Volterra's and Lokshin's theory in studying well-posedness and regularity of solutions to wave equations; examining step signal propagation in hereditary medium; formulating and analyzing the fundamental solution to the anisotropic multi-dimensional linear viscoelastic wave equation; formulating Hamiltonian and Lagrangian theory of viscoelasticity; formulating restrictions that allow the complete monotonicity; general stability result; and analysis of the Zener fractional wave equation, respectively. Wave equation obtained in modelling string vibration is analyzed in [60,65].…”

Time domain characterization of the Cole-Cole dielectric model

“…Its permittivity is " r @!A a " I C " s " I I C j! ; (10) where " s is the static value and " I < " s is the value at innity frequency, is a characteristic time constant for the medium. The rst term, represented by the constant " I , represents an ideal capacitor which is in parallel with a frequency-varying part.…”

“…It is given by ©@tA a p I I "@!A [5]. In this way the physical realizability of the considered system is guaranteed [10].…”

“…This is what characterizes a completely monotone function and it ensures that it can be expressed as a continuous distribution of exponential functions and that the Laplace transform is non-negative [ 5 ]. In this way the physical realizability of the considered system is guaranteed [ 10 ].…”